Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-6: Replace the symbol \(\square\) with either \(<,>\), or \(=\) to make the resulting statement true. (a) \(\frac{1}{7} \square 0.143\) (b) \(\frac{5}{6} \square 0.833\) (c) \(\sqrt{2} \square 1.4\)

Step-by-Step Solution

Answer

(a) <; (b) >; (c) >

Step 1: Calculate \\ \( \frac{1}{7} \)

Let's calculate the decimal equivalent of \( \frac{1}{7} \). Divide 1 by 7 to get the decimal: \( \frac{1}{7} \approx 0.142857 \). This is a repeating decimal.

Step 2: Compare Decimals for Part (a)

We need to compare the decimal \( 0.142857 \) with \( 0.143 \). Since \( 0.142857 < 0.143 \), the correct symbol to use is \( < \). So, \( \frac{1}{7} < 0.143 \).

Step 3: Calculate \\ \( \frac{5}{6} \)

Determine the decimal value of \( \frac{5}{6} \). Divide 5 by 6 to get \( \frac{5}{6} \approx 0.8333 \).

Step 4: Compare Decimals for Part (b)

Compare the decimal \( 0.8333 \) with \( 0.833 \). Since \( 0.8333 > 0.833 \), the correct symbol is \( > \). So, \( \frac{5}{6} > 0.833 \).

Step 5: Calculate \( \sqrt{2} \)

The square root of 2 is approximately \( 1.414 \).

Step 6: Compare \\ \( \sqrt{2} \) with \( 1.4 \)

Compare \( 1.414 \) with \( 1.4 \). Since \( 1.414 > 1.4 \), the correct symbol is \( > \). So, \( \sqrt{2} > 1.4 \).

Key Concepts

Decimal Conversion

Understanding decimal conversion is crucial when dealing with fractions. Often, fractions need to be converted into decimals to facilitate easier comparison or computation.

• The process involves dividing the numerator by the denominator. For example, to convert \( \frac{1}{7} \) to a decimal, divide 1 by 7, which gives approximately 0.142857, a repeating decimal.
• Recognize repeating decimals, which have a digit or group of digits that repeat indefinitely after the decimal point.
• Knowing how to interpret and use these repeating decimals is important as they often appear in math problems.

Regular practice with conversion can greatly enhance computational efficiency in various mathematical exercises. Pay close attention to the decimal place values, as these will dictate comparisons between numbers.

Square Roots

Square roots reveal the number which, when multiplied by itself, gives the original number. They are prevalent in many mathematical concepts and require accurate approximation skills.

• The square root of a number is represented by the radical symbol \( \sqrt{} \). For instance, \( \sqrt{2} \) is approximately 1.414.
• Understanding square roots aids in solving quadratic equations and plays a role in geometry and other advanced math concepts.
• Rounding the square root to a certain number of decimal places is crucial when comparing with other numbers. This ensures accuracy in inequality comparisons.

While some square roots are integers (perfect squares), many, like \( \sqrt{2} \), are irrational numbers which continue indefinitely without repeating.

Algebraic Symbols

Algebraic symbols such as \(<\), \(>\), and \(=\) are used to express relationships between numbers, especially in inequalities and equations.

• The symbol \(<\) means "less than," indicating that the number on the left is smaller than the number on the right, as seen when \( \frac{1}{7} < 0.143 \).
• The symbol \(>\) denotes "greater than," showing the left side is larger, just like \( \frac{5}{6} > 0.833 \).
• The equality symbol \(=\) signifies that two values are identical, although it wasn't used in the given comparisons.

Comprehending these symbols is critical as they form the basis of comparing values, solving equations, and analyzing functions in algebra.